» Khovanskii A.N. - The application of continued fractions
Khovanskii A.N. - The application of continued fractions
||The application of continued fractions
||In modern mathematics the approximate representation of functions is ordinarily sought for in the form of a polynomial in the independent variable. In cases in which such polynomials are difficult to find, other numerical methods are used.
For this purpose approximations by rational functions of the independent variable have seldom been used. A characteristic of rational function approximations is that they may often successfully represent the given function in a domain of variation of the argument where the power series expansion of the function diverges and where, in consequence, in a great number of cases a polynomial approximation is inapplicable.
Furthermore, with the help of rational function approximations, the determination of the zeros and poles of the given function is greatly facilitated, since it is required to solve an algebraic equation of lower degree than that which occurs when using an approximation in the form of a polynomial.
Finally, the use of a rational function approximation tends to remove the necessity of computing high powers of the argument.
Thus the application of rational function approximations brings about a great simplification in many of the computing formulae.
That approximation by means of a rational function should have gained so small a currency is explained by the fact that the direct derivation of this function necessitates lengthy calculations. Furthermore the transition from one rational function approximation to another involves, in general, the recomputation of all coefficients contained in the numerators and denominators of these approximations. However methods exist allowing the derivation of arbitrarily many rational function approximations to the given function, in a manner not demanding complicated calculations. The most widely known methods of this type are based on the use of continued fractions.